The sensitivity analysis of hypergame equilibrium

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1 3rd International Conference on Manageent, Education, Inforation and Control (MEICI 015) The sensitivity analysis of hypergae equilibriu Zhongfu Qin 1,a Xianrong Wei 1,b Jingping Li 1,c 1 College of Civil Engineering and Architecture Zheiang University,Hangzhou,China a qinzhongfu@zu.edu.cn, b @qq.co, c Keywords: Hypergae; Equilibriu; AHP; Sensitivity analysis Abstract. Hypergae is that people calculate the equilibriu solution of the conflict by udging other players preference vector according to their own subective opinion. But in the case of hypergae, to obtain a correct opponent players preference inforation is very iportant. In this paper, AHP and sensitivity analysis are used to solve this proble. Turn a proble into ultiple conflict analysis probles, then get each equilibriu of the conflict analysis. In this process the player s cognition to other players can be ore accurate. The player can get ore stronger stability of equilibriu. Introduction Strategy selection of conflict analysis is based on that the inforation is transparent to each other, and each player is rational and obective. However, the real conflict is not always in the case, inforation distortion and players error udgent are frequent. For this kind of situation, hypergae technology becoes an effective decision analysis technology. Hypergae is the conflict that one or ore players didn't realize the nature of the conflict. It is put forward in the 1970 s by the western scholars Bennett (1977) [1]. Suppose the ethod: Due to that the player has a wrong order of other players preferences or isunderstand others available strategy, etc., he has a deviation on decision situation []. Under the hypergae, getting the proper opponent players preference inforation is very iportant. Song Y, (008) [3] develops an iterative algorith to integrate the coprehensive results of rival preference inforation. Because it s difficulty in the process of hypergae to percept the players preference perception quantitatively. Song Y, [4] ( 009) presents equilibriu results obtained ethod which have a fuzzy strategies and preference cognitive. For a certain credibility of decision aker, deterine the new strategy set and preference recognition results according to the fuzzy cut and the distance of fuzzy nuber. In order to reduce the extent of the gae in isunderstanding, haresifard B, s Cortes. [5] (010) think people can obtain experience by observing the behavior of other players to odify their perspective of other players payoff function (preferences), the authors introduce a isunderstanding function to easure the differences between player s view and other players real payoff function. In order to discuss the stability of hypergae, Sasaki y. [7] (008), defines a new solution concept that we call the Nash hyper equilibriu stability. This concept solves the stability proble that after each hypergae, isunderstanding retain in the gae. Moreover, haresifard B, Cortes.[6](011), designed a higher-order cognitive update algorith which allow players continue to update their perception, assuing that players are rational and have perfect observation to the result of the gae, 015. The authors - Published by Atlantis Press 694

2 and use this inforation to update their preference perception of rivals, so any repeated hypergae will converge to an equilibriu. This paper, based on the above scholars research, firstly, finds out the influence factors of hypergae, then conducts sensitivity analysis on the influencing factors. Thus, it builds ultiple conflict analysis odels, and analyzes the change and stability of the equilibriu solution. Part introduces the hypergae equilibriu odel, and Part 3 introduces the sensitivity analysis ethod of hypergae. Equilibriu odel of Hypergae In the stability analysis, hypergae odel can be converted into several siple conflict analysis odel [8]. In this paper, we analysis the proble fro the perspective of the player I. If I s preferences is stable, player J s preferences which player I think is also stable, then, I s equilibriu solution in the gae is indeed the equilibriu can solve the conflict which player I think[9.10]. As shown in Table1, Player I thinks Player J has preference vectors, Player I s sensitivity analysis is i the corresponding gae is that Player I cognitives gae i. This article does not consider Players J istake cognition, naely Player J have no isunderstanding to Player I. Player J can correctly perceive Player I, so every tie Player J s gae is only. Considering the whole hypergae, ( i, ) will get the ultiate equilibriu solution. Players' cognition Player I cognizes preference of ector I Table 1 ii Two-person gae in hypergae Players' perception of gae ii... ii Player I cognizes preference of ector J 1 i i... i The gae Player I cognizes 1 i i i Player J cognizes preference of ector I i i... i Player J cognizes preference of ector J... The gae Player J cognizes... Total gae, ) ( i, ) ( 1 i..., ) ( i 695

3 Sensitivity Analysis of equilibriu solution for hypergae The order of the initial preference vector of player i to player by chroatography analysis. Assuing in a hypergae Player I thinks that there are factors(evaluation standards) influence its udgent about Player J s preference vector sequence, as shown in Fig. 1. And each evaluation standard has a certain weight value, respectively, f1,f,...,f, f 1 1. Using analytic hierarchy process(ahp) ethod to calculate the weight, then the several evaluation criteria for pairwise coparison, copared to the values own for doinance ( /3 1/5 1/7 1/9), according to the advantage, calculate weight values. Then all the outcoes (assuing has K outcoes) are of pairwise coparison under the each standard influence. With the ethod of AHP, calculate its weight value under each evaluation criterion. Finally, weight of influence factors and outcoe scores are weighted suation, find out the final score ik of each end of the cognition of Player I k C f 1 Finally, sort the preference vector, according to the calculated value fro big to sall are arranged, the initial preference vector k 0 i of player i to player. Situation evaluation of player i to player Ultiate goal Evaluation criteria 1 Evaluation criteria Evaluation criteria.. Evaluation criteria Evaluation criteria Situation1 Situation Situation 3 Situation 4 Situation k Alternative Fig. 1 Hierarchical analysis diagra Sensitivity analysis on the cognitive preference vector of Player I to Player J. The doinance change of evaluation standards for pairwise coparison, hange at a close of each value for a tie, were changed to n ties, n = 1,,..., n n=; Calculate the weight values fn of the evaluation criteria after each change. etting the final score of each outcoe of the player i : C f ; To sort n, get the preference vector i ik 1 k n ( sorted fro high to low); Draw preference vector sensitivity analysis chart, statistics preference vectors type and nuber; 696

4 Find equilibriu solution fro n kinds of preference vectors. Analyze the equilibriu solution fro Player I and equilibriu solution of total proble. Discussion and suary Calculate the equilibriu solution under each preference vector, and statistic the types and the nuber of equilibriu solution, and copare the with the initial equilibriu of Player I. If consistent, suggesting that Player I s prediction of the equilibriu is relatively stable, and ay be easy to appear in the real world. The player I can stick to its current udgent on other players preference vector. To note here, it can stick to this kind of udgent, which doesn't ean it can think of this kind of udgent is right, because of what it ows, even if the current preference vector udgent of other players and real preference vector are difference, it also does not affect its rightly prediction of the equilibriu. If not consistent, Player I should change the preferences vector views to J. Should Player I choose the equilibriu solution to appeare the ost frequently? The frequency highest equilibriu is not necessarily in the ost satisfactory solution, it is ust the ost stable solution. This solution is likely to let one player feel very unsatisfactory. Soeties even let all the parties feel unhappy. Second, though the equilibriu solution is the ost stable solution, the reality of the final outcoe of the dispute is not necessary in this solution. For exaple, Player I thinks Player I has A, B, C, D, E possible preference vectors, but it cannot confir the true preference vector. If A, B and C preference vectors equilibriu solution is in situation 1, under D and E preference vector,the equilibriu solutions are in situation and situation 3, but the real preference vector of player J in the bureau is D. So this case seeks equilibriu solution is a little coplex, whether Player I thinks the initial preference vector and the highest nuber of preferences vector are probably the real preference vector of Player I in bureau, but it ight not be. How to deal with this situation, the author has not studied in this part. This paper has not discussed in detail. But the author will discuss this question in the future. References [1]N. M. Fraser and K. W. Hipel, Conflict Analysis: Models and Resolutions [D]. Systes Science and Engineering, Oxford, UK: Elsevier, [].,. Benntt: Bidders and dispenser: anipulative hypergaes in a ultinational context. European Journal of Operational Research 4 (1980) [3]Xiu Y, Song Y, Zhang J. An Outcoe preference Inforation Aggregation Model and Its Algorith in Hypergae Situations[C]//Fuzzy Systes and Knowledge Discovery, 008. FSKD'08. Fifth International Conference on. IEEE, 008, 4: [4]Song Y, Li W, Zeng X.λ-equilibriu of hypergae with fuzzy strategy and preference perceptions[c]//intelligent Coputing and Intelligent Systes, 009. ICIS 009. IEEE International Conference on. IEEE, 009, : [5]haresifard B, Cortés J. Evolution of the perception about the opponent in hypergaes[c]//decision and Control (CDC), th IEEE Conference on. IEEE, 010:

5 [6]haresifard B, Cortés J. Learning of equilibria and isperceptions in hypergaes with perfect observations[c]//aerican Control Conference (ACC), 011. IEEE, 011: [7]Sasaki Y. preservation of Misperceptions Stability Analysis of Hypergaes[C]//proceedings of the 5nd Annual Meeting of the ISSS-008, Madison, Wisconsin. 008, 3(1). [8] Lu Xiangna, conflict analysis, a new decision-aking ethod [J], ournal of anageent engineering,1993, [9]Yong L B S Y Q. Interactive Integration Model of Hypergaes with Fuzzy Strategy and Preference Perceptions [J]. Coputer & Digital Engineering, 011, : 00. [10]ane R, Lehner P. Using hypergaes to increase planned payoff and reduce risk[j]. Autonoous Agents and Multi-Agent Systes, 00, 5(3):

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